Submitted by Petr Vesely on

Founded 15-May-2006

Last update 15-May-2006

Antioch Mint, 1st Reign Antioch Mint, 3rd Reign References

## Antioch Mint, 1st Reign

### 1. Examined type

Denomination: |
AR Tetradrachm |

Period: |
121/0 - spring/summer 113 BC |

Obverse: |
Diademed head of Antiochos VIII right; fillet border |

Reverse: |
‘ΒΑΣΙΛΕΩΣ ΑΝΤΙΟΧΟΥ’ right, ‘ΕΠΙΦΑΝΟΥΣ’ left; Zeus Ouranios standing left, crescent above head, holding star in outstretched right hand and scepter with left arm; ‘ΙΕ’ monogram above ‘Α’ in outer left field; laurel wreath border |

### 2. Acceptable weight range

Lower exclusion limit: |
15.50 grams |

Upper exclusion limit: |
17.00 grams |

Each coin is a priori excluded from the data sample if its weight is lesser than the lower exclusion limit or greater than the upper exclusion limit.

### 3. Data

Sorted data (weights in grams):

15.99, 16.16, 16.20, 16.22, 16.23, 16.27, 16.35, 16.35, 16.35, 16.37, 16.38, 16.42, 16.42, 16.43, 16.44, 16.46, 16.47, 16.47, 16.48, 16.48, 16.48, 16.52, 16.53, 16.53, 16.54, 16.54, 16.54, 16.56, 16.59, 16.60, 16.61, 16.61, 16.62, 16.66, 16.68, 16.73, 16.74, 16.82

**Note:** The following coins were included into the analysis:

- Auktionshaus Meister & Sonntag: Auction 2 (Sep 2004), Lot 1020
- Calgary Coin and Antique: fixed price list (recorded May 2006)
- CIVITAS Galleries, Ltd.: fixed price list (recorded May 2006)
- Classical Numismatic Group, Inc.: Mail Bid Sale 60 (May 2002), Lots 926 and 927; Triton VI (Jan 2003), Lot 458; Mail Bid Sale 64 (Sep 2003), Lot 402; Mail Bid Sale 66 (May 2004), Lot 693; Mail Bid Sale 67 (Sep 2004), Lot 891; Triton VIII (Jan 2005), Lot 549; Mail Bid Sale 70 (Sep 2005), Lots 385 - 387
- Dr. Busso Peus Nachf.: Auction 376 (Oct 2003), Lot 533
- Fritz Rudolf Künker Müzenhandlung: Auction 71 (Mar 2002), Lot 436; Auction 77 (Sep 2002); Lot 243; Auction 83 (Jun 2003), Lot 416; Auction 89 (Mar 2004), Lot 1478; Auction 97 (Mar 2005), Lots 932 - 934; Auction 111 (Mar 2006), Lot 6316
- John C. Lavender, Classical Numismatist: fixed price list (recorded May 2006)
- Hess-Divo AG: Auction 299 (Oct 2004), Lot 106
- Houghton, CSE, 322
- Numismatik Lanz München: Auction 112 (Nov 2002), Lot 240; Auction 125 (Nov 2005), Lot 476
- Pegasi Numismatics: fixed price list (recorded May 2006)
- SNG Spaer, 2493, 2494, 2496 - 2500
- Sphinx Numismatics: fixed price list (recorded May 2006)
- The British Sylloge Nummorum Graecorum: Vol. IV 5787 Fitzwilliam Museum (SNGuk_0408_5787); Vol: IV 5788 Fitzwilliam Museum (SNGuk_0408_5788)

### 4. Descriptive statistics

No. of observations: |
38 | |

Mean: |
16.47 | (95% confidence interval: 16.41 ≤ mean ≤ 16.53) |

Standard deviation: |
0.17 | |

Interquartile range: |
0.22 | |

Skewness: |
-0.49 | |

Kurtosis: |
3.38 | |

Minimum: |
15.99 | |

25th percentile: |
16.37 | (96.1% confidence interval: 16.23 ≤ 25th percentile ≤ 16.46) |

Median: |
16.48 | (96.6% confidence interval: 16.42 ≤ median ≤ 16.54) |

75th percentile: |
16.59 | (96.1% confidence interval: 16.53 ≤ 75th percentile ≤ 16.66) |

Maximum: |
16.82 |

**Notes:** The unbiased estimation of the variance was used for the computation of the standard deviation (i.e. the number of observations minus one was used as a divisor). The sample skewness was computed without sample corrections (i.e. the skewness was computed as the square root of the number of observations times the sum of the third powers of deviations from the mean divided by the 3/2 power of the sum of the squares of deviations from the mean). Similarly, the sample kurtosis was computed as the number of observations times the sum of the fourth powers of deviations from the mean divided by the second power of the sum of the squares of deviations from the mean.

The confidence interval for mean was computed by using the Student t-distribution. The confidence intervals for median and percentiles were computed nonparametrically by using the binomial distribution (see, e.g., Conover, *Practical Nonparametric Statistics*, pp. 143 - 148).

### 5. Estimation of proportion of coins with weights within the observed range

At the 95% level of confidence, at least 88.1% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 15.99 g and 16.82 g, and at least 80.8% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 16.16 g and 16.74 g.

**Note:** These estimations are computed as tolerance limits based on the binomial distribution. See, e.g., Conover, *Practical Nonparametric Statistics*, pp. 150 - 155.

### 6. Histogram and probability density function

Histogram of the sample is presented in Figure 1. Kernel estimations of the probability density function are shown in Figure 2 (two kernel estimations were used: Epanechnikov kernel with a bandwidth of 0.062 and Gaussian kernel with a bandwidth of 0.071). The dotted curve in Figure 2 is a probability density function of a normal distribution estimated by the maximum likelihood method.

**Note:** The bandwidth of the Gaussian kernel was computed as h_{Gauss} = 0.9 × min(σ, SIQR) × n^{-1/5}, where σ is the standard deviation, SIQR is the standardised interquartile range (i.e. the interquartile range of the data sample divided by the interquartile range of the standard normal density) and n is the number of observations; see Silverman, *Density Estimation for Statistics and Data Analysis*, p. 48, formula (3.31). The bandwidth of the Epanechnikov kernel was chosen subjectively in the range from 0.75×h_{Gauss} to 1.25×h_{Gauss}.

### 7. Test of normality

The Lilliefors test of normality was used. The test statistic of 0.097 is less than the cutoff value of 0.144 for a 95% level test. Thus we cannot reject the hypothesis of normality at the 95% level of significance. Normal probability plot of the sample is presented in Figure 3.

## Antioch Mint, 3rd Reign

### 1. Examined type

Denomination: |
AR Tetradrachm |

Period: |
109 - 96 BC |

Obverse: |
Diademed head of Antiochos VIII right; fillet border |

Reverse: |
‘ΒΑΣΙΛΕΩΣ ΑΝΤΙΟΧΟΥ’ right, ‘ΕΠΙΦΑΝΟΥΣ’ left; Zeus Nikephoros seated left on throne, holding Nike in outstretched right hand and scepter in left hand; laurel wreath border |

### 2. Acceptable weight range

Lower exclusion limit: |
15.00 grams |

Upper exclusion limit: |
17.00 grams |

Each coin is a priori excluded from the data sample if its weight is lesser than the lower exclusion limit or greater than the upper exclusion limit.

### 3. Data

Sorted data (weights in grams):

15.58, 15.61, 15.62, 15.64, 15.66, 15.79, 15.80, 15.84, 15.88, 15.88, 15.92, 15.95, 15.97, 15.98, 15.98, 16.00, 16.02, 16.04, 16.08, 16.09, 16.11, 16.12, 16.14, 16.15, 16.15, 16.16, 16.16, 16.20, 16.21, 16.21, 16.24, 16.24, 16.25, 16.30, 16.31, 16.32, 16.33, 16.35, 16.36, 16.37, 16.38, 16.47, 16.49

**Note 1:** The following coins were included into the analysis:<

- Classical Numismatic Group, Inc.: Mail Bid Sale 58 (Sep 2001), Lot 708; Mail Bid Sale 63 (May 2003), Lot 661; Mail Bid Sale 67 (Sep 2004), Lots 893 and 894; Electronic Auction 104 (Dec 2004), Lot 107; Triton VIII (Jan 2005), Lot 550; Electronic Auction 107 (Feb 2005), Lot 97; Electronic Auction 109 (Mar 2005), Lot 55
- Dr. Busso Peus Nachf.: Auction 368 (Apr 2001), Lot 293; Auction 376 (Oct 2003), Lot 535
- Frank Sternberg AG: Auction 23 (Oct 2000), Lot 307
- Fritz Rudolf Kuenker Muenzenhandlung: Auction 67 (Oct 2001), Lot 455; Auction 77 (Sep 2002), Lot 246; Auction 89 (Mar 2004), Lot 1480; Auction 94 (Sep 2004), Lot 1430
- Glenn W. Woods, Numismatist: fixed price list (recorded May 2006)
- Gorny & Mosch Giessener Münzhandlung: Auction 108 (Apr 2001), Lot 1363; Auction 118 (Oct 2002), Lot 1521; Auction 130 (Mar 2004), Lot 1293; Auction 134 (Oct 2004), Lot 1559; Auction 142 (Oct 2005), Lot 1650
- Hess-Divo AG: Auction 299 (Oct 2004), Lot 107
- Harlan J. Berk, Ltd.: fixed price list (recorded May 2006)
- Houghton, CSE, 346 - 348
- LHS Numismatik AG: Auction 95 (Oct 2005), Lot 717
- Numismatik Lanz München: Auction 97 (May 2000), Lot 343; Auction 120 (May 2004), Lot 175
- SNG Spaer, 2554 - 2558
- The British Sylloge Nummorum Graecorum: Vol. I 441 Newnham Davis Coins (SNGuk_0102_0441); Vol. III 3176 Lockett Collection (SNGuk_0300_3176); Vol. III 3177 Lockett Collection (SNGuk_0300_3177); Vol. IV 5795 Fitzwilliam Museum (SNGuk_0408_5795); Vol. IV 5796 Fitzwilliam Museum (SNGuk_0408_5796); Vol. IV 5797 Fitzwilliam Museum (SNGuk_0408_5797); Vol. VI 1100 Fitzwilliam Museum (SNGuk_0601_1100); VII 1356 Manchester University Museum (SNGuk_0700_1356); VII 1357 Manchester University Museum (SNGuk_0700_1357)

**Note 2:** The following coin was not included because its weight exceeds the upper exclusion limit: The British Sylloge Nummorum Graecorum, Vol. I 443 Newnham Davis Coins (SNGuk_0102_0443), weight 17.01 g.

### 4. Descriptive statistics

No. of observations: |
43 | |

Mean: |
16.08 | (95% confidence interval: 16.00 ≤ mean ≤ 16.15) |

Standard deviation: |
0.24 | |

Interquartile range: |
0.32 | |

Skewness: |
-0.44 | |

2.39 | ||

Minimum: |
15.58 | |

25th percentile: |
15.93 | (94.9% confidence interval: 15.79 ≤ 25th percentile ≤ 16.02) |

Median: |
16.12 | (93.4% confidence interval: 16.00 ≤ median ≤ 16.20) |

75th percentile: |
16.25 | (94.9% confidence interval: 16.16 ≤ 75th percentile ≤ 16.35) |

Maximum: |
16.49 |

**Notes:** The unbiased estimation of the variance was used for the computation of the standard deviation (i.e. the number of observations minus one was used as a divisor). The sample skewness was computed without sample corrections (i.e. the skewness was computed as the square root of the number of observations times the sum of the third powers of deviations from the mean divided by the 3/2 power of the sum of the squares of deviations from the mean). Similarly, the sample kurtosis was computed as the number of observations times the sum of the fourth powers of deviations from the mean divided by the second power of the sum of the squares of deviations from the mean.

The confidence interval for mean was computed by using the Student t-distribution. The confidence intervals for median and percentiles were computed nonparametrically by using the binomial distribution (see, e.g., Conover, *Practical Nonparametric Statistics*, pp. 143 - 148).

### 5. Estimation of proportion of coins with weights within the observed range

At the 95% level of confidence, at least 89.4% of issued coins of the examined type have a weight between the smallest observation and the largest observation, i.e. between 15.58 g and 16.49 g, and at least 82.9% of issued coins of the examined type have a weight between the second smallest observation and the second largest observation, i.e. between 15.61 g and 16.47 g.

**Note:** These estimations are computed as tolerance limits based on the binomial distribution. See, e.g., Conover, *Practical Nonparametric Statistics*, pp. 150 - 155.

### 6. Histogram and probability density function

Histogram of the sample is presented in Figure 4. Kernel estimations of the probability density function are shown in Figure 5 (two kernel estimations were used: Epanechnikov kernel with a bandwidth of 0.096 and Gaussian kernel with a bandwidth of 0.101). The dotted curve in Figure 5 is a probability density function of a normal distribution estimated by the maximum likelihood method.

**Note:** The bandwidth of the Gaussian kernel was computed as h_{Gauss} = 0.9 × min(σ, SIQR) × n^{-1/5}, where σ is the standard deviation, SIQR is the standardised interquartile range (i.e. the interquartile range of the data sample divided by the interquartile range of the standard normal density) and n is the number of observations; see Silverman, *Density Estimation for Statistics and Data Analysis*, p. 48, formula (3.31). The bandwidth of the Epanechnikov kernel was chosen subjectively in the range from 0.75×h_{Gauss} to 1.25×h_{Gauss}.

### 7. Test of normality

The Lilliefors test of normality was used. The test statistic of 0.090 is less than the cutoff value of 0.135 for a 95% level test. Thus we cannot reject the hypothesis of normality at the 95% level of significance. Normal probability plot of the sample is presented in Figure 6.

### References:

**Conover, W. J.:***Practical Nonparametric Statistics*, Third Edition. John Wiley & Sons, Inc., New York - Chichester - Weinheim - Brisbane - Singapore - Toronto, 1999.**Houghton, Arthur:***Coins of the Seleucid Empire from the Collection of Arthur Houghton*. The American Numismatic Society, New York, 1983. (abbr.*CSE*)**Houghton, Arthur; Spaer, Arnold (with the assistance of Catharine Lorber):***Sylloge Nummorum Graecorum. Israel I. The Arnold Spaer Collection of Seleucid Coins*. Italo Vecchi Ltd., London, 1998. (abbr. SNG*Spaer*)**Silverman, B.W.:***Density Estimation for Statistics and Data Analysis*. Chapman and Hall, London - New York - Tokyo - Melbourne - Madras, 1993 (reprint of the first edition published in 1986).